Problem 66 - Discrete Mathematics with Applications

Question

Exercises \(65-78\) deal with propositions in fuzzy logic. Let \(p, q,\) and \(r\) be simple propositions with \(t(p)=1, t(q)=0.3,\) and \(t(r)=\) 0.5 . Compute the truth value of each, where \(s^{\prime}\) denotes the negation of the statement \(s\) . $$ p \wedge q $$

Step-by-Step Solution

Answer

The truth value of the compound proposition \(p \wedge q\) is \(0.3\).

Step 1: Identify Given Truth Values

We are given the truth values for the simple propositions: \(t(p)=1\), \(t(q)=0.3\), and \(t(r)=0.5\)

Step 2: Apply Conjunction Rule For Fuzzy Logic

To find the truth value of the compound proposition \(p \wedge q\), we need to use the conjunction rule in fuzzy logic, which states that the truth value of a conjunction of two propositions is the minimum of their truth values.

Step 3: Compute The Truth Value Of The Compound Proposition

Using the conjunction rule, we have: \(t(p\wedge q) = min(t(p), t(q)) = min(1, 0.3) = 0.3\) So, the truth value of the compound proposition \(p \wedge q\) is \(0.3\).